How to modify width of "wire" material type
Apart the Z offset setting shown in another answer, there are at least other 2 ways to render that wireframe: freestyle and wireframe modifier.

The first draws stylized lines over the resulting object after all rendering finish. You can customize the lines styles and where to apply them, like:



The second method can be used with two copies of the object: one gets the usual colored material, the other has the wireframe modifier, and gets another material just for the wire mesh to display, and you have some degree of editing for  the wireframe "style"



here you see how the wireframe is really a separate object, I scaled it up a bit...